This question just popped up again from an edit and I was rather surprised that, considering the amount of interest, no one has suggested a rather obvious integration theory that also answers the question.
We want a "teaching integral" that is adequate to get the students started on a study of integration theory but not overwhelm them with the correct modern theories. Some do suggest the Henstock-Kurzweil integral. This makes sense only if the students have already studied the rather pathetic Riemann integral. We say, then, "Here is a better theory, no harder than what you just learned, and much more powerful if you care to examine it further." But the Henstock-Kurzweil integral is a solution to a different problem. It is not "integration with training wheels." It is a full theory of integration that includes the Riemann integral, the Lebesgue integral, the improper Riemann integral, and the Denjoy-Perron integral.
Some of us feel that a return to the Newton integral is a better introduction to integration theory as a first step. After all, this is the way that every 18th century mathematician viewed integration theory.
Definition A function $f:[a,b]\to\mathbb R$ is Newton integrable on $[a,b]$ if there is an antiderivative $F$ (i.e., $F'(x)=f(x)$ for
all $a\leq x \leq b$) and then one defines $$\int_a^b
The justification for the integral is simply the mean-value of the calculus. The basic properties of the integral are all deduced from properties of derivatives. Riemann sums come in by way of the mean-value theorem too.
I think if you did a poll of calculus students who have been force-fed the Riemann integral, nearly all of them would say that this is indeed all that they consider integration to be. It is how they all calculate integrals --although they remember vaguely some unpleasant times spent using Riemann sums, but mercifully no-one makes them do that stuff any more.
Does this integral handle all continuous functions? You can do exactly as Cauchy did and show how to construct the primitive of continuous functions using Riemann sums. I prefer myself to prove this lemma:
Lemma Let $f:[a,b]\to\mathbb R$ be a bounded function. Then there is a Lipschitz function $F:[a,b]\to\mathbb R$ so that
$F'(x)=f(x)$ at every point $x$ at which $f$ is continuous.
It is not that hard and can be presented at an elementary level if you don't demand the students follow in detail every step. You don't need uniform continuity evidently, but you do need to construct a sequence of functions that converges to the function you want.
For more information about how one might possibly take this point of view for an introductory course in integration theory you can consult this experimental textbook.